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The stabilization of homogeneous cubic vector fields in the plane
0893-9659(94)00059-X
Appl. Math. Lett. Vol. 7, No. 4, pp. 95-99, 1994 Copyright@1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0893-9659/94 $7.00 + 0.00
The Stabilization of Homogeneous Cubic Vector Fields in the Plane M. A. HAMMAMI AND H. JERBI INRIA-Lorraine (CONGE Project) & URA CNRS 399 (MMAS) CESCOM Technopole Metz-2000. 4, rue Marconi, 57070 Metz France {hammami,jerbi}Bilm.loria.fr (Received October 1993; accepted November 1993) Abstract-In this paper, we deal with the stabilizability of homogeneous polynomials of degree three by means of homogeneous feedbacks of degree two.
Keywords-Homogeneous polynomials of degree three, Nonlinear systems, Homogeneous feedbacks of degree two, Stabilizability.
1. INTRODUCTION In this paper, we study the stabilizability
problem
E =
for planar nonlinear
P(x) + UBX
systems of the form
(1.1)
(T = where x = (xl,x~)~ E lR2, u E R, B is a 2 x 2 matrix and P(x) = (Pl(x),P~(x))~, transpose) with PI, P2 homogeneous polynomials of degree three. Such a kind of problem has been addressed by authors like [1,2], where in [l], they consider systems of the form ? = f(x) +Bu, where f is a homogeneous vector fields of odd degree. The authors gave a sufficient condition for global stabilizability by means of homogeneous feedbacks of the same degree. The difference with our problem is that the controlled vector field is more degenerated; this is in fact the main difficulty. In this work, we give a sufficient condition for the stabilizability of the system (1.1) by homogeneous feedbacks of degree two. Our method consists to transform the system into the form ? = Q(z)Az
+ i.iBx
O-2)
where A is a constant matrix and Q(x) is a positive definite quadratic form on R2. Therefore, the stabilizability of the system can be deduced thanks to the classification of planar bilinear systems ? = Aa:+ uBx, given by [3], where for stabilizable bilinear systems, a smooth on lR2 \ (0) positively homogeneous of degree zero feedback is given. Throughout this paper, we shall consider the function F(x) = det(P(x), Bz). It turns out that, if F(x) has a quadratic positive definite factor Q(x), by using some changes of feedbacks, one can transform the system (1.1) into the form (1.2). Furthermore, as an application of this study, by using the original system, we give a result on the local stabilization of nonlinear systems affine in control on RF. Typeset by &S-T@ 95
M. A. HAMMAMIAND H. JERBI
96
2. STABILIZABILITY Consider the system (l.l),
where B = (bl(~),bz(~))~
with bl(zl,zz) = bllzl + blsxz and bz(xl,z~) = bzlxl +bzzxz. Our goal is to seek a (2 x 2) constant matrix A, Ax = (~~(x),a~(x))~, and a feedback u = U(X) such that 51 jl, 0
_ 9(x1,22) - ( pz(x1,~2)
bl(xl,x2)
= Q(x1,x2)
bz(xl,zz)
(;;If::3
where &I is a positive definite quadratic form. We wish, by making a change of feedback u = u(x1, ~2) + ii, to write the system (1.1) as (1.2), because we have the following result: THEOREM
1. The two following statements are equivalents.
1. The system i = Q(x)Ax + iiBx is globally asymptotically stable. 2. The planar bilinear system 5 = Ax + uBx is globally asymptotically stable. PROOF.
(l)-+(2): It is shown in [3] that the stabilizability of planar bilinear systems where the control u E R, is equivalent to the asymptotic controllability to the origin. Therefore, if one suppose that the bilinear system is not G.A.S, then it can not be asymptotically controllable to the origin. Since the form quadratic Q is positive definite, it follows that the system (1.2) is not asymptotically controllable to the origin, so it can not be G.A.S. (2)-+(l): If the planar bilinear system ? = Ax + uBx is G.A.S, then it is G.A.S by u = u(x) homogeneous feedback law of degree zero. Since Q(x) is a quadratic positive definite form, it implies that the system 2 = Q(x) (Ax + u(x)Bx) is G.A.S, and so the system (1.2) is G.A.S by ii = Q(x)u(x) homogeneous feedback of degree two. In [3], the authors gave a classification of planar bilinear systems which are stabilizable by homogeneous feedbacks of degree zero. So to say that system ? = Q(x)Ax + iiBx is G.A.S by homogeneous feedback of degree 2, is equivalent to say that the system k = Ax +uBx is G.A.S by homogeneous feedback of degree zero. Now, in order to transform the system (1.1) if possible, we shall use the function F. The polynomial F has three forms. F is the product of two quadratics definite, either one quadratic definite and two linears, or F takes the form of four linears product. However, by using the canonical forms of a given, degree four homogeneous polynomial, (see [4, Theorem 2.61). One can easily put F in one of the three forms given above. These forms should be treated separately. 2.1. Suppose that F(xl, x2) = Ql(x1, x2) . &2(x1, x2). Without loss of generality, we can suppose that Q1 is positive definite quadratic form and Q2 is a definite quadratic form. 1. There exists a constant matrix A and a change of feedback u = ~(21, x2) + 6 where u(x1, x2) is homogeneous of degree two, which transform the system (1.1) into the system
PROPOSITION
(1.2). PROOF.
Since F(xI,x~)
is definite, then
1 0
= U
Ql(ar x2)
( $z::::;
it j::: ::i)
-l (::[z::
::;)
The latter in conjunction with the form of F, yields Q2(21,~2)
= al(xl,x2)b2(xl,x2)
-a2(~1,xz)bl(xl,x2)
(2.1)
and u(x17x2) =
Q2(x;,x2,
(-q(21,22)P2(x1,
x2) + a2(x1, X2)Pl(Xl,
z2)).
(2.2)
97
Homogeneous Cubic Vector Fields
On the one hand, one can write (2.1) as zT
(k
z::)
Ax = Q2(x).
Since &Z(X) is definite
quadratic form, then there exists a matrix definite 0, such that Q~(z) = z’&z.
Hence, by using
the fact that det B # 0 because F is supposed a definite function, it follows that, one choice of the constant matrix A, which is a solution of the equation given above, is
On the other hand, one can verify that the function ~(21, ~2) given by (2.2), is homogeneous of degree two. 2.2. Suppose now that F(x1,22) = QI(zI,~~)(~I~I + P1x2)(azxl + Pzzz), where m7~2,P1,P2 and &1(21,~2) = & + ~2122 + cxz, such that v E lR, p > 0, [ > 0, which satisfies Y2 - 4& > 0. E IR
PROPOSITION
2.
There exist some changes of feedback such that, in a suitable basis, the system
(1.1) can be written as (1.2). PROOF. CASE
We deal with the two following situations for the matrix B. 1. det B # 0. In this case, the two vectors line bl and b2 are independents.
Letting
F(xl,x2) = b2(xl,x2)Pl(xl,x2) L~(x~,xz) = ~~1x1+ ,%x2 and Lz(x~,zz) = ~2x1 +&.x2. bl(xl,x2)P2(x1,x2) = Q1(xl,x~)~L~(x~,x~)~L~(xl,x~). SincedetB #O, byusingsomechanges of coordinates, one can suppose that bll # 0. Thus, we consider the Eucldean division on R[(z~,x~)] of PI by bl and the product L2.Ql by bl. Therefore, there exist some quadratic
forms 41, q2 and some constants 11, 12 such that PI(x~,x~) = ql(x1,x2)bl(x1,x2) + lox: and 3 The term xg appears in the last decomposition =q2(xl,x~)bl(xl,x2)+~2~2.
~2(51,x)Q1(~1,~2)
because bll is supposed nonzero. Now, if 11 # 0, then the two polynomials PI and bl are relatively prime (r.p.), and hence bl and F are also r.p. Since L2 . &I divide F, it follows that L2 . &I and bl are r.p., and so 12 # 0. Thus, PI = 11//2(L~.Ql - qzbl) + qlbl. Hence, under a change in the input space of the form ‘u.-+ 11/12q2(xlrx2) - ql(x1, x2) + u, the system (1.1) becomes k, = 5L2(xl,~2) 12
k2 = 4(51,x2)
.Q1(~1,~2)
+ubl(~l,~2),
+ ub2(Zlrx2),
(2.3)
where k2 = P2 + ((11/h)q2 - ql)b2. Since F is invariant by change of feedback, then L2.Q1 divide 4 so, there exists a constant k such that, &(x1, x2) = kL2(x1, x2) . Ql(xl, x2). One gets
0 kl
= &1(a:1,~2).
i2
A(::>
+uB(;:)
where A=
If 11 = 0, then the system (2.3) can be written after a change of feedback of the form u = Ql(Xl,
x2) + ii, as kl 0 k2
=&1(x174
(L2cx;,x2j) +~(;$3.
CASE 2. det B = 0. Setting P(x, t) = tlx: + t2xTx2 + tsx~xij + tdx& ti E JR. In a suitable basis, Jordan basis, the matrix B can be written as . In this case, let PI(x~,x~) = P(x,s) and P~(x~,Ic~) = P(x,r). One can > verify that F(xI, x2) = -X1x1 (TIX~ + 73x:x2 + ~3212%+ ~42:). Taking into account the
(1) B =
(
^d
;
M. A. HAMMAMI AND H. JERBI
98
form of F, one can choose crl = -Xl u = Exixz 01
+ 2x;
and /31 = 0. Then, with the change of feedback
+ 6, one has
01
On this basis, letting Pi = P(x, s’) and P2 = P(z,T'), thenusingthesame I St< idea as above, the change of feedback u = - (shxf + six1x2 + six;) +S’Vx:+1xix2+~, P P makes the system (1.1) as follows
REMARK. Suppose that F(x) takes the form of four linears product. In this case it is not possible to transform the system (1.1)into the system (1.2), because if it exists a such transformation, then necessarily Q(x) divide F(x), which is not the case. So the method seen in 2.1 and 2.2 cannot be applicable in this case where F(x) has not a quadratic definite factor. In the following section, we now present a result on the local stabilization of nonlinear systems in dimensional n, to illustrate the applicability of our main result.
3. LOCAL
STABILIZATION
OF NONLINEAR
SYSTEMS
We consider a single input nonlinear system, defined on a neighborhood U of the origin of Rn x = f(x) + ug(x).
(3.1)
We suppose that f E C3 and g E C1 functions such that f(0) = O,g(O) = 0. DEFINITION. A function cp is said to be positively homogeneous of degree m 2 0,if for any vector x and any positive real k, one has cp(kx) = km(p(x). Assume that, Dfc = 0 and D2fa = 0, since f(0) = 0 and g(0) = 0, one can write the vector fields f and g as, f(x) = D3f ox3 + fl(x) and g(x) = Dgcx + gr(x) where fi and gi satisfies 3 and ]]gi(x)]/ < M~]]x]], Vx E U’ c U. Denoting P for D3fo and B for Dgo, Ilf1(x)Il 5 ~lll4l let us consider the system i = P(x) + uBx. (3.2) We shall call (3.2), the approximating system for the system (3.1). THEOREM 2. If the approximating system (3.2) is stabilizable by means of positively homogeneous feedback of degree two, then the system (3.1) is locally stabilizable. PROOF. Let u = u(x) be a positively homogeneous stabilizing feedback of degree two for the approximating system (3.2). Set v(x) = P(x) + u(x)Bx and 4(x) = fi(x) + u(x)g(x). Since u is C’ on V’\(O), it can be seen that cp and 4 are locally lipschitz. Besides, one has ]u(x)] 5 M~]]x]]~. Furthermore, one can verify that cp is positively homogeneous of degree three and 4 satisfies ]]4(x)]] I (M~+MoM~)]]x]]~, Vx E U’ thus, ]]+(x)]] I M]]x]]3, Vx E U’, where M = MI +MoM2. Then it follows from [5, Theorem 241, that the solution x = 0 of the equation x = v(x) + 4(x) is asymptotically stable. Hence, the system (3.1) is locally asymptotically stable. EXAMPLE.
Consider the following system ii = sinxi3 - 5xTxz + 2xi(cosxz x2 = -xi
- 1)2 + 9x; + usinxi,
+ 2x:x2 + 3(e”’ - 1) x2 + xi + ~~0~x2 - 21.
(3.3)
Cubic Vector Fields
Homogeneous
99
Then, the approximating system for the system (3.3) is il = Lr; - 52& f2 = -x;
+ 2x&
+ 9x; + t&El,
+ 2x722 + 3x12; + x; - ux2.
(3.4)
A simple computation gives F(xl, x2) = (2; - 21x2 + 3xg)(xl - 3x2)(21 +x2).
Here Q(xl, 52) =
x9 - 1~1x2+ 3x:. If one considers the following change of feedback u = 22: + 5x1~2 + 4x;, +ti, then the system (3.4) becomes
0q:=
(XT-
x,x~+~x~)A(::)+BB(:?),
whereA=
(_y
_f),
andB=
(i
_y).
-12x:
stabilizes the bilinear 3x7 +xcg’ system ? = Ax + iiBx, where x = (xl, x2). Thus, by application of Theorem 1, one can deduce
From [2], the homogeneous feedback of degree zero 6(x1, x2) = that the homogeneous feedback of degree two -12x! U(Xl,X2)
stabilizes the system (3.4). stabilizable.
=
+ 12x;x2 - 36x:x; 32: + x;
Hence, it follows by Theorem 2, that the system (3.3) is locally
REFERENCES 1. A. Andreini, A. Bacciotti and G. Stefani, Global stabilizability of homogeneous vector fields of odd degree, Systems and Control Letters 10, 251-256, (1988). 2. A. Baccioti and P. Boieri, A characterization of single input planar bilinear systems which admit a smooth stabilizer, Systems and Control Letters 16, 139-143, (1991). 3. R. Chabour, G. Sallet and J.C. Vivalda, Stabilization of nonlinear two dimensional systems: A bilinear approach, In NOLCOS’SB, Bordeaux, France, June 24-26, 1992. 4. A. Cima and J. Llibre, Algebric and topological classification of the homogeneous cubic vector fields in the plane, Journal of Mathematical Analysis and Applications 14, 420-448, (1990). 5. J.L. Massera, Contribution to stability theory, Annals of Mathematics 64, 182-206, (1956).
Appl. Math. Lett. Vol. 7, No. 4, pp. 95-99, 1994 Copyright@1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0893-9659/94 $7.00 + 0.00
The Stabilization of Homogeneous Cubic Vector Fields in the Plane M. A. HAMMAMI AND H. JERBI INRIA-Lorraine (CONGE Project) & URA CNRS 399 (MMAS) CESCOM Technopole Metz-2000. 4, rue Marconi, 57070 Metz France {hammami,jerbi}Bilm.loria.fr (Received October 1993; accepted November 1993) Abstract-In this paper, we deal with the stabilizability of homogeneous polynomials of degree three by means of homogeneous feedbacks of degree two.
Keywords-Homogeneous polynomials of degree three, Nonlinear systems, Homogeneous feedbacks of degree two, Stabilizability.
1. INTRODUCTION In this paper, we study the stabilizability
problem
E =
for planar nonlinear
P(x) + UBX
systems of the form
(1.1)
(T = where x = (xl,x~)~ E lR2, u E R, B is a 2 x 2 matrix and P(x) = (Pl(x),P~(x))~, transpose) with PI, P2 homogeneous polynomials of degree three. Such a kind of problem has been addressed by authors like [1,2], where in [l], they consider systems of the form ? = f(x) +Bu, where f is a homogeneous vector fields of odd degree. The authors gave a sufficient condition for global stabilizability by means of homogeneous feedbacks of the same degree. The difference with our problem is that the controlled vector field is more degenerated; this is in fact the main difficulty. In this work, we give a sufficient condition for the stabilizability of the system (1.1) by homogeneous feedbacks of degree two. Our method consists to transform the system into the form ? = Q(z)Az
+ i.iBx
O-2)
where A is a constant matrix and Q(x) is a positive definite quadratic form on R2. Therefore, the stabilizability of the system can be deduced thanks to the classification of planar bilinear systems ? = Aa:+ uBx, given by [3], where for stabilizable bilinear systems, a smooth on lR2 \ (0) positively homogeneous of degree zero feedback is given. Throughout this paper, we shall consider the function F(x) = det(P(x), Bz). It turns out that, if F(x) has a quadratic positive definite factor Q(x), by using some changes of feedbacks, one can transform the system (1.1) into the form (1.2). Furthermore, as an application of this study, by using the original system, we give a result on the local stabilization of nonlinear systems affine in control on RF. Typeset by &S-T@ 95
M. A. HAMMAMIAND H. JERBI
96
2. STABILIZABILITY Consider the system (l.l),
where B = (bl(~),bz(~))~
with bl(zl,zz) = bllzl + blsxz and bz(xl,z~) = bzlxl +bzzxz. Our goal is to seek a (2 x 2) constant matrix A, Ax = (~~(x),a~(x))~, and a feedback u = U(X) such that 51 jl, 0
_ 9(x1,22) - ( pz(x1,~2)
bl(xl,x2)
= Q(x1,x2)
bz(xl,zz)
(;;If::3
where &I is a positive definite quadratic form. We wish, by making a change of feedback u = u(x1, ~2) + ii, to write the system (1.1) as (1.2), because we have the following result: THEOREM
1. The two following statements are equivalents.
1. The system i = Q(x)Ax + iiBx is globally asymptotically stable. 2. The planar bilinear system 5 = Ax + uBx is globally asymptotically stable. PROOF.
(l)-+(2): It is shown in [3] that the stabilizability of planar bilinear systems where the control u E R, is equivalent to the asymptotic controllability to the origin. Therefore, if one suppose that the bilinear system is not G.A.S, then it can not be asymptotically controllable to the origin. Since the form quadratic Q is positive definite, it follows that the system (1.2) is not asymptotically controllable to the origin, so it can not be G.A.S. (2)-+(l): If the planar bilinear system ? = Ax + uBx is G.A.S, then it is G.A.S by u = u(x) homogeneous feedback law of degree zero. Since Q(x) is a quadratic positive definite form, it implies that the system 2 = Q(x) (Ax + u(x)Bx) is G.A.S, and so the system (1.2) is G.A.S by ii = Q(x)u(x) homogeneous feedback of degree two. In [3], the authors gave a classification of planar bilinear systems which are stabilizable by homogeneous feedbacks of degree zero. So to say that system ? = Q(x)Ax + iiBx is G.A.S by homogeneous feedback of degree 2, is equivalent to say that the system k = Ax +uBx is G.A.S by homogeneous feedback of degree zero. Now, in order to transform the system (1.1) if possible, we shall use the function F. The polynomial F has three forms. F is the product of two quadratics definite, either one quadratic definite and two linears, or F takes the form of four linears product. However, by using the canonical forms of a given, degree four homogeneous polynomial, (see [4, Theorem 2.61). One can easily put F in one of the three forms given above. These forms should be treated separately. 2.1. Suppose that F(xl, x2) = Ql(x1, x2) . &2(x1, x2). Without loss of generality, we can suppose that Q1 is positive definite quadratic form and Q2 is a definite quadratic form. 1. There exists a constant matrix A and a change of feedback u = ~(21, x2) + 6 where u(x1, x2) is homogeneous of degree two, which transform the system (1.1) into the system
PROPOSITION
(1.2). PROOF.
Since F(xI,x~)
is definite, then
1 0
= U
Ql(ar x2)
( $z::::;
it j::: ::i)
-l (::[z::
::;)
The latter in conjunction with the form of F, yields Q2(21,~2)
= al(xl,x2)b2(xl,x2)
-a2(~1,xz)bl(xl,x2)
(2.1)
and u(x17x2) =
Q2(x;,x2,
(-q(21,22)P2(x1,
x2) + a2(x1, X2)Pl(Xl,
z2)).
(2.2)
97
Homogeneous Cubic Vector Fields
On the one hand, one can write (2.1) as zT
(k
z::)
Ax = Q2(x).
Since &Z(X) is definite
quadratic form, then there exists a matrix definite 0, such that Q~(z) = z’&z.
Hence, by using
the fact that det B # 0 because F is supposed a definite function, it follows that, one choice of the constant matrix A, which is a solution of the equation given above, is
On the other hand, one can verify that the function ~(21, ~2) given by (2.2), is homogeneous of degree two. 2.2. Suppose now that F(x1,22) = QI(zI,~~)(~I~I + P1x2)(azxl + Pzzz), where m7~2,P1,P2 and &1(21,~2) = & + ~2122 + cxz, such that v E lR, p > 0, [ > 0, which satisfies Y2 - 4& > 0. E IR
PROPOSITION
2.
There exist some changes of feedback such that, in a suitable basis, the system
(1.1) can be written as (1.2). PROOF. CASE
We deal with the two following situations for the matrix B. 1. det B # 0. In this case, the two vectors line bl and b2 are independents.
Letting
F(xl,x2) = b2(xl,x2)Pl(xl,x2) L~(x~,xz) = ~~1x1+ ,%x2 and Lz(x~,zz) = ~2x1 +&.x2. bl(xl,x2)P2(x1,x2) = Q1(xl,x~)~L~(x~,x~)~L~(xl,x~). SincedetB #O, byusingsomechanges of coordinates, one can suppose that bll # 0. Thus, we consider the Eucldean division on R[(z~,x~)] of PI by bl and the product L2.Ql by bl. Therefore, there exist some quadratic
forms 41, q2 and some constants 11, 12 such that PI(x~,x~) = ql(x1,x2)bl(x1,x2) + lox: and 3 The term xg appears in the last decomposition =q2(xl,x~)bl(xl,x2)+~2~2.
~2(51,x)Q1(~1,~2)
because bll is supposed nonzero. Now, if 11 # 0, then the two polynomials PI and bl are relatively prime (r.p.), and hence bl and F are also r.p. Since L2 . &I divide F, it follows that L2 . &I and bl are r.p., and so 12 # 0. Thus, PI = 11//2(L~.Ql - qzbl) + qlbl. Hence, under a change in the input space of the form ‘u.-+ 11/12q2(xlrx2) - ql(x1, x2) + u, the system (1.1) becomes k, = 5L2(xl,~2) 12
k2 = 4(51,x2)
.Q1(~1,~2)
+ubl(~l,~2),
+ ub2(Zlrx2),
(2.3)
where k2 = P2 + ((11/h)q2 - ql)b2. Since F is invariant by change of feedback, then L2.Q1 divide 4 so, there exists a constant k such that, &(x1, x2) = kL2(x1, x2) . Ql(xl, x2). One gets
0 kl
= &1(a:1,~2).
i2
A(::>
+uB(;:)
where A=
If 11 = 0, then the system (2.3) can be written after a change of feedback of the form u = Ql(Xl,
x2) + ii, as kl 0 k2
=&1(x174
(L2cx;,x2j) +~(;$3.
CASE 2. det B = 0. Setting P(x, t) = tlx: + t2xTx2 + tsx~xij + tdx& ti E JR. In a suitable basis, Jordan basis, the matrix B can be written as . In this case, let PI(x~,x~) = P(x,s) and P~(x~,Ic~) = P(x,r). One can > verify that F(xI, x2) = -X1x1 (TIX~ + 73x:x2 + ~3212%+ ~42:). Taking into account the
(1) B =
(
^d
;
M. A. HAMMAMI AND H. JERBI
98
form of F, one can choose crl = -Xl u = Exixz 01
+ 2x;
and /31 = 0. Then, with the change of feedback
+ 6, one has
01
On this basis, letting Pi = P(x, s’) and P2 = P(z,T'), thenusingthesame I St< idea as above, the change of feedback u = - (shxf + six1x2 + six;) +S’Vx:+1xix2+~, P P makes the system (1.1) as follows
REMARK. Suppose that F(x) takes the form of four linears product. In this case it is not possible to transform the system (1.1)into the system (1.2), because if it exists a such transformation, then necessarily Q(x) divide F(x), which is not the case. So the method seen in 2.1 and 2.2 cannot be applicable in this case where F(x) has not a quadratic definite factor. In the following section, we now present a result on the local stabilization of nonlinear systems in dimensional n, to illustrate the applicability of our main result.
3. LOCAL
STABILIZATION
OF NONLINEAR
SYSTEMS
We consider a single input nonlinear system, defined on a neighborhood U of the origin of Rn x = f(x) + ug(x).
(3.1)
We suppose that f E C3 and g E C1 functions such that f(0) = O,g(O) = 0. DEFINITION. A function cp is said to be positively homogeneous of degree m 2 0,if for any vector x and any positive real k, one has cp(kx) = km(p(x). Assume that, Dfc = 0 and D2fa = 0, since f(0) = 0 and g(0) = 0, one can write the vector fields f and g as, f(x) = D3f ox3 + fl(x) and g(x) = Dgcx + gr(x) where fi and gi satisfies 3 and ]]gi(x)]/ < M~]]x]], Vx E U’ c U. Denoting P for D3fo and B for Dgo, Ilf1(x)Il 5 ~lll4l let us consider the system i = P(x) + uBx. (3.2) We shall call (3.2), the approximating system for the system (3.1). THEOREM 2. If the approximating system (3.2) is stabilizable by means of positively homogeneous feedback of degree two, then the system (3.1) is locally stabilizable. PROOF. Let u = u(x) be a positively homogeneous stabilizing feedback of degree two for the approximating system (3.2). Set v(x) = P(x) + u(x)Bx and 4(x) = fi(x) + u(x)g(x). Since u is C’ on V’\(O), it can be seen that cp and 4 are locally lipschitz. Besides, one has ]u(x)] 5 M~]]x]]~. Furthermore, one can verify that cp is positively homogeneous of degree three and 4 satisfies ]]4(x)]] I (M~+MoM~)]]x]]~, Vx E U’ thus, ]]+(x)]] I M]]x]]3, Vx E U’, where M = MI +MoM2. Then it follows from [5, Theorem 241, that the solution x = 0 of the equation x = v(x) + 4(x) is asymptotically stable. Hence, the system (3.1) is locally asymptotically stable. EXAMPLE.
Consider the following system ii = sinxi3 - 5xTxz + 2xi(cosxz x2 = -xi
- 1)2 + 9x; + usinxi,
+ 2x:x2 + 3(e”’ - 1) x2 + xi + ~~0~x2 - 21.
(3.3)
Cubic Vector Fields
Homogeneous
99
Then, the approximating system for the system (3.3) is il = Lr; - 52& f2 = -x;
+ 2x&
+ 9x; + t&El,
+ 2x722 + 3x12; + x; - ux2.
(3.4)
A simple computation gives F(xl, x2) = (2; - 21x2 + 3xg)(xl - 3x2)(21 +x2).
Here Q(xl, 52) =
x9 - 1~1x2+ 3x:. If one considers the following change of feedback u = 22: + 5x1~2 + 4x;, +ti, then the system (3.4) becomes
0q:=
(XT-
x,x~+~x~)A(::)+BB(:?),
whereA=
(_y
_f),
andB=
(i
_y).
-12x:
stabilizes the bilinear 3x7 +xcg’ system ? = Ax + iiBx, where x = (xl, x2). Thus, by application of Theorem 1, one can deduce
From [2], the homogeneous feedback of degree zero 6(x1, x2) = that the homogeneous feedback of degree two -12x! U(Xl,X2)
stabilizes the system (3.4). stabilizable.
=
+ 12x;x2 - 36x:x; 32: + x;
Hence, it follows by Theorem 2, that the system (3.3) is locally
REFERENCES 1. A. Andreini, A. Bacciotti and G. Stefani, Global stabilizability of homogeneous vector fields of odd degree, Systems and Control Letters 10, 251-256, (1988). 2. A. Baccioti and P. Boieri, A characterization of single input planar bilinear systems which admit a smooth stabilizer, Systems and Control Letters 16, 139-143, (1991). 3. R. Chabour, G. Sallet and J.C. Vivalda, Stabilization of nonlinear two dimensional systems: A bilinear approach, In NOLCOS’SB, Bordeaux, France, June 24-26, 1992. 4. A. Cima and J. Llibre, Algebric and topological classification of the homogeneous cubic vector fields in the plane, Journal of Mathematical Analysis and Applications 14, 420-448, (1990). 5. J.L. Massera, Contribution to stability theory, Annals of Mathematics 64, 182-206, (1956).